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Quadratic formula and factoring patterns

23 formulas · what each one means · a free lesson for each

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Quadratic equations

Standard form
ax2 + bx + c = 0
Everything on one side, 0 on the other, before you solve.Lesson: Quadratic equations →
Quadratic formula
x = −b ± √b2 − 4ac2a
Works for every quadratic equation.Lesson: Quadratic equations →
Discriminant
b2 − 4ac
Positive: two answers. Zero: one answer. Negative: no real answers.Lesson: The discriminant →
Zero product rule
ab = 0 → a = 0 or b = 0
Why factoring solves: one of the brackets must be 0.Lesson: Solving by factoring →
Square root way
x2 = k → x = ±√k
When there is no x term.Lesson: Quadratic equations →
Completing the square
x2 + bx + (b2)2 = (x + b2)2
Add half of b, squared, to make a perfect square.Lesson: Completing the square →
Sum and product of the roots
r1 + r2 = −ba r1 × r2 = ca
Read from the equation without solving.Lesson: Sum and product of the roots →
Thrown ball (feet)
h = −16t2 + v0 t + h0
v₀ = starting speed, h₀ = starting height. In meters: −5t² (−4.9t²).Lesson: Quadratic word problems →

Parabolas

Vertex (x part)
x = −b2a
The middle of the parabola; put it in to get y.Lesson: Parabolas →
Vertex form
y = a(x − h)2 + k
Vertex (h, k). a > 0 opens up, a < 0 opens down.Lesson: Vertex form by completing the square →
Axis of symmetry
x = h
The line through the vertex that folds the parabola in half.Lesson: Parabolas →

Factoring patterns

Common factor
ab + ac = a(b + c)
Take out what every term shares.Lesson: Factoring →
x² + bx + c
x2 + bx + c = (x + p)(x + q), p × q = c, p + q = b
Find two numbers that multiply to c and add to b.Lesson: Factoring →
ax² + bx + c (split)
two numbers: product ac, sum b
Split bx with them, then factor by grouping.Lesson: Factoring →
Grouping
ax + ay + bx + by = (a + b)(x + y)
Factor each pair, then take out the shared bracket.Lesson: Factor by grouping →
Difference of squares
a2 − b2 = (a + b)(a − b)
Two squares with a minus between them.Lesson: Factoring →
Perfect square
a2 + 2ab + b2 = (a + b)2
First and last are squares, middle is 2 × a × b.Lesson: Factoring completely →
Sum of cubes
a3 + b3 = (a + b)(a2 − ab + b2)
Lesson: Sum and difference of cubes →
Difference of cubes
a3 − b3 = (a − b)(a2 + ab + b2)
Signs: same, opposite, always plus.Lesson: Sum and difference of cubes →

Multiplying brackets

Two brackets
(a + b)(c + d) = ac + ad + bc + bd
Each part of the first bracket times each part of the second.Lesson: Polynomials →
Square of a sum
(a + b)2 = a2 + 2ab + b2
Not a² + b²: the middle 2ab is easy to forget.Lesson: Polynomials →
Square of a difference
(a − b)2 = a2 − 2ab + b2
The middle part is minus, the last part is plus.Lesson: Polynomials →
Sum times difference
(a + b)(a − b) = a2 − b2
The middle parts cancel.Lesson: Polynomials →

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